determination of rolling-element fatigue life from ... of rolling-element fatigue life ... virtual...

Brian L. VlcekGeorgia Southern University, Statesboro, Georgia

Robert C. Hendricks and Erwin V. ZaretskyGlenn Research Center, Cleveland, Ohio

Determination of Rolling-Element Fatigue LifeFrom Computer Generated Bearing Tests

NASA/TM—2003-212186

August 2003

https://ntrs.nasa.gov/search.jsp?R=20030093532 2018-05-31T00:05:34+00:00Z

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Brian L. VlcekGeorgia Southern University, Statesboro, Georgia

Robert C. Hendricks and Erwin V. ZaretskyGlenn Research Center, Cleveland, Ohio

Determination of Rolling-Element Fatigue LifeFrom Computer Generated Bearing Tests

NASA/TM—2003-212186

August 2003

National Aeronautics andSpace Administration

Glenn Research Center

Prepared for the2003 Annual Meeting and Exhibitionsponsored by the Society of Tribologists and Lubrication EngineersNew York City, New York, April 28–May 1, 2003

Acknowledgments

The authors would like to acknowledge the contribution of Gregory Hickman of Georgia Southern University forthe generation of many spreadsheets used in the Monte Carlo analysis in this paper.

Available from

NASA Center for Aerospace Information7121 Standard DriveHanover, MD 21076

National Technical Information Service5285 Port Royal RoadSpringfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov

NASA/TM�2003-212186 1

DETERMINATION OF ROLLING-ELEMENT FATIGUE LIFE FROM COMPUTER GENERATED BEARING TESTS

Brian L. Vlcek Georgia Southern University

Statesboro, Georgia 30460�8045

Robert C. Hendricks and Erwin V. Zaretsky National Aeronautics and Space Administration

Glenn Research Center Cleveland, Ohio 44135

SUMMARY

Two types of rolling-element bearings representing radial loaded and thrust loaded bearings were used for this study. Three hundred forty (340) virtual bearing sets totaling 31400 bearings were randomly assembled and tested by Monte Carlo (random) number generation. The Monte Carlo results were compared with endurance data from 51 bearing sets comprising 5321 bearings. A simple algebraic relation was established for the upper and lower L10 life limits as function of number of bearings failed for any bearing geometry. There is a fifty percent (50%) probability that the resultant bearing life will be less than that calculated. The maximum and minimum variation between the bearing resultant life and the calculated life correlate with the 90-percent confidence limits for a Weibull slope of 1.5. The calculated lives for bearings using a load-life exponent p of 4 for ball bearings and 5 for roller bearings correlated with the Monte Carlo generated bearing lives and the bearing data. STLE life factors for bearing steel and processing provide a reasonable accounting for differences between bearing life data and calculated life. Variations in Weibull slope from the Monte Carlo testing and bearing data correlated. There was excellent agreement between percent of individual components failed from Monte Carlo simulation and that predicted.

NOMENCLATURE

C, CD dynamic load capacity, N (lbf) c stress-life exponent e Weibull slope h exponent L life, number of stress cycles or hr L10 10-percent life or life at which 90 percent of a population survives, number of stress cycles or hr Lβ characteristic life or life at which 63.2 percent of population fails, number of stress cycles or hr N life, number of stress cycles n number of failed bearings or number of elements in population P, Peq equivalent radial load, N (lbf) p load-life exponent S probability of survival, fraction or percent V stressed volume, m3, (in.3) Zo depth to the orthogonal shearing stress, m (in.) ττττo orthogonal shearing stress, GPa (ksi) Subscripts i ith component or bearing ir inner race L lower limit max maximum min minimum n number of components or outer race

NASA/TM�2003-212186 2

re rolling elements sys system up upper limit Definitions calculated life the life obtained using the Lundberg-Palmgren life equations resultant life the life obtained from the Weibull analysis of bearing systems generated by a Monte Carlo technique

INTRODUCTION

Predicting and verifying rolling-element bearing life is a complex task. Accurate prediction of bearing lives is necessary to predict replacement rates, maintain rotating machinery and establish warranty limits on manufactured goods. Complicating the issue is the fact that fatigue failure is extremely variable and dependent upon materials, processing, and operating conditions.

Rolling-bearing fatigue life analysis is based on the initiation or first evidence of fatigue spalling on either a bearing race or a rolling element (ball or roller). This spalling phenomenon is load cycle dependent. Generally, the spall begins in the region of maximum shear stresses, which is located below the contact surface, and propagates into a crack network. Failures other than those caused by classical rolling-element fatigue are considered avoidable if the bearing is properly designed, handled, installed, lubricated and not overloaded (1). However, under low elastohydrodynamic (EHD) lubricant film conditions, rolling-element fatigue can be surface or near-surface initiated with the spall propagating into the region of maximum shearing stresses.

If a number of apparently identical bearings are tested to fatigue at a specific load, there is a wide dispersion in life among the various bearings. For a group of 30 or more bearings the ratio of the longest to the shortest life may be 20 or more (1).

In 1939, Weibull (2-4) developed a method and an equation for statistically evaluating the fracture strength of materials based upon small population sizes. This method can be and has been applied to analyze, determine, and predict the cumulative statistical distribution of fatigue failure or any other phenomenon or physical characteristic that manifests a statistical distribution.

Based upon the work of Weibull (2), Lundberg, and Palmgren (5), in 1947, showed that the probability of survival S could be expressed as a power function of the orthogonal shear stress ττττo, life N, depth to the maximum orthogonal shear stress Zo, and stressed volume V. That is

VZ

N

S ho

e

oττττ~1

n1 (1)

From Eq. (1), Lundberg and Palmgren (5) derived the following relation

L10 = [CD /Peq]p (2)

where CD, the basic dynamic load capacity, is defined as the load that a bearing can carry for one million inner-race revolutions with a 90-percent probability of survival, Peq is the equivalent bearing load, and p is the load life exponent. The derivation of Eq. (2) is discussed in Zaretsky et al (6).

The term �basic rating life,� as used in bearing catalogs, usually means the fatigue life exceeded by 90 percent of the bearings or the time before which 10 percent of the bearings fail. This basic rating life is referred to as the �L10 life� (sometimes called the B10 life or 10-percent life). The 10-percent life is approximately one-seventh of the mean life or MTBF (mean time between failure), for a bearing life dispersion curve (1).

Harris (7,8) analyzed 62 rolling-element bearing endurance sets. These data were obtained from four bearing manufacturers, two helicopter manufacturers, three aircraft engine manufacturers, and U.S. Government agency-sponsored technical reports. The data sets comprised deep-groove radial ball bearings, angular-contact ball bearings, and cylindrical roller bearings totaling 7935 bearings.

Using the Harris data (7,8), Zaretsky, Poplawski, and Miller (9) compared the ratio of the L10 lives of the field and laboratory bearing life data to that predicted by various life theories discussed in Ref. (6). For the Lundberg-Palmgren equations discussed above, the mean ratios of the L10 actual lives divided by the L10 predicted lives (determined using STLE life factors (1)) were 14.5, 3.5, and 20.1 for angular-ball bearings, deep-groove ball bearings, and cylindrical roller bearings, respectively. While it is probable that all design and operating parameters

NASA/TM�2003-212186 3

necessary to more accurately calculate bearing life were not available to Harris (7,8), these bearing data are the best compilation in the open literature.

An issue that occurs from these data and analysis is the variation between bearing life calculations and the actual endurance characteristics of the bearings. Experience has shown that endurance tests of groups of identical bearings under identical conditions can produce variation in L10 life from group to group that exceeds reasonable engineering expectations, that is, where the life is significantly less or more than that calculated. This is an important issue for product warranty, comparison of bearings from different sources and variation in life from lot-to-lot from the same source.

In view of the aforementioned, and using the Lundberg-Palmgren analysis (5) as the basis, the objectives of the work reported herein are: (a) to determine the variation in rolling-element bearing lives and distribution parameters as a function of sample size (number of bearings tested); (b) compare the statistical variation in bearing life due to finite sample size to the Harris rolling-element bearing data (7,8); and (c) determine the most likely value of the load-life exponent based upon a comparison of the field data of Harris (7,8) to the upper and lower limits (or 90% confidence limits) obtained for a Weibull-based Monte Carlo prediction of bearing life.

PROCEDURE

Bearing Life Analysis

G. Lundberg and A. Palmgren (5), using the Weibull equation (2�4), first derived the relationship between

individual component lives and system life where

[ ] [ ]∑=

=n

i

eesys

1

/1/1 iLL (3)

Using Eq. (1), Lundberg and Palmgren (5) develop equations for the lives of the inner and outer races of a

bearing and combine them using Eq. (3) to determine the bearing life at a 10-percent probability of failure or the time beyond which 90 percent of the bearings will survive where

[1/L10]

e = [1/Lir]e + [1/Lor]

e (4) Unfortunately, Lundberg and Palmgren (5) do not directly calculate the lives of the rolling element (ball or

rollers) set of the bearing. However, through the benchmarking of the equations with bearing life data by use of a material-geometry factor, the life of the rolling elements are implicitly included in the life calculation of Eq. (4).

The rationale for not including the rolling elements in Eq. (4) appears in the 1945 edition of A. Palmgren�s book (10) wherein he states that, ��the fatigue phenomenon which determines the life (of the bearing) usually develops on the raceway of one ring or the other. Thus, the rolling elements are not the weakest parts of the bearing � .� The data base that Palmgren used to benchmark his and later the Lundberg-Palmgren equations were obtained under radially loaded conditions. Under these conditions the life of the rolling elements as a system will be equal or greater than the outer race. As a result, failure of the rolling elements in determining bearing life was not initially a consideration by Palmgren. Equation (4) should be written as follows

[1/L10]

e = [1/Lir]e + [1/Lre]

e + [1/Lor]e (5)

where the Weibull slope e is the same for each of the components as well as the bearing as a system.

Comparing Eq. (5) with Eq. (4), the value of the L10 bearing life will be the same. However, the values of the Lir and Lor between the two equations will not be the same, but, the ratio of Lor/Lir will remain unchanged.

In order to account for material and processing variations between the rolling elements and the races, it is important to break out the ball or roller life from that of the inner and outer races using Eq. (5). This can be accomplished using Zaretsky�s Rule (1) as follows

For radially loaded ball and roller bearings, the life of the rolling element set is equal to or greater than the life of the outer race. Let the life of the rolling element set (as a system) be equal to that of the outer race.

NASA/TM�2003-212186 4

From Eq. (5)

[1/L10]e = [1/Lir]

e + 2[1/Lor]e (6)

where Lre = Lor

For thrust loaded ball and roller bearings, the life of the rolling element set is equal to or greater than the life of the inner race but less than that of the outer race. Let the life of the rolling element set (as a system) be equal to that of the inner race. From Eq. (5)

[1/L10]e = 2[1/Lir]

e + [1/Lor]e (7)

where Lre = Lir

Examples for using Eqs. (5) to (7) are given in Zaretsky (1). As previously stated, the resulting values for Lir

and Lor from these equations are not the same as those from Eq. (4).

Bearing Type, Operating Conditions and Calculated Lives

Two types of rolling-element bearings representing radial loaded and thrust loaded bearings were used for this study. They were a 6010-size (50-mm bore) deep-groove ball bearing and a 7010-size (50-mm bore) angular-contact ball bearing, respectively. The bearing specifications and geometry are summarized in Table 1. For purposes of this analysis all life factors such as for material and processing were set to unity since we were interested primarily in the qualitative results. However, a lubricant life factor was used as a function of lubricant film parameter from Zaretsky (1) for these operating conditions since its effect on the resulting lives of the inner and outer races can be different.

Table 1. Bearing specifications, operating conditions, and lives used in

assembly and Monte Carlo testing Bearing type Deep-groove

ball bearing Angular-contact ball

bearing Bore size, mm 50 50

Inner race 52 52 Curvatures, percent Outer race 52 52 Ball diameter, mm (in.) 8.73 (11/32) 8.73 (11/32) Number of balls 14 19 Contact angle, deg 0 25 Load, N (lbs) 950 (214) radial 2800 (630) thrust Maximum Hertz stress, GPa (ksi) 1.55 (225) 1.55 (225) Lubricant type MIL�L�23699 MIL�L�23699

Inner race 7.62×10�2 (3) 7.62×10�2 (3) Outer race 7.62×10�2 (3) 7.62×10�2 (3)

Surface finish, rms µ (µm)

Balls 2.54×10�2 (1) 2.54×10�2 (1) Operating temperature, ºC (ºF) 135 (275) 135 (275)

Inner race 0.75 0.79 Lubricant life factors Outer race 1.05 1.04

Component L10 L50 L10 L50 Inner racea 9547 52123 1974 10775 Outer racea 38188 208448 7885 43040 Balla 38118 208448 1974 10775

Life, hrs (see Fig. 1)

Bearingb 6912 37729 964 5262 Weibull slope, e 1.11 1.11

aLife based on Zaretsky�s rule and lubricant life factor (from Ref. (1)). bLife based on Lundberg-Palmgren equations (from Ref. (5)) and lubricant life factor (from Ref. (1)).

NASA/TM�2003-212186 5

Operating conditions for both bearing types were assumed to be 10000 rpm using a MIL�L�23699 (tetraester based) lubricant at 135 °C (275 °F). The respective loads applied to both bearings were calculated to result in a maximum Hertz stress on the inner race of each bearing of 1.55 GPa (225 ksi). These operating conditions are summarized in Table 1.

The bearing lives were calculated according to Lundberg-Palmgren Eqs. (1) and (4) with a lubricant life factor. The lives of the inner and outer races were calculated together with the lives of the balls using Eqs. (6) and (7). These results are also summarized in Table 1. Weibull plots of the bearing and their individual component lives are shown in Fig. 1.

Virtual Bearing Testing

A rolling-element bearing is composed of

4 components. These are the inner and outer races and a plurality of rolling elements that are positioned and retained by a separator (cage) between the two races. The life of a bearing is probabilistic and is calculated based upon rolling-element fatigue (spalling failure) of either the inner or outer races and/or the rolling elements. Upon the formation of a spall on anyone of these components, the bearing is no longer fit for its intended purpose and is subject to being replaced. The separator is assumed not to fail under normal operating conditions. The variables that affect bearing life are discussed in detail in Ref. (1).

The cumulative distribution of the individual components of the two bearing types is shown in the two-parameter Weibull plots of Fig. 1. The general equation representing these plots is as follows

10;0 whereln 1

lnln <<∞<<

=

βSL

L

Le

S (8)

The Weibull plots shown in Fig. 1 are the S

1lnln graduated in percent of bearings or components failed as the

ordinate as a function of ln L, the log of the time or cycles to failure as the abscissa. The tangent of the line is designated as the Weibull slope e. The Weibull slope e is indicative of the shape of the cumulative distribution of the data. Based upon their database, Lundberg and Palmgren (5) use a value for the Weibull slope e of 1.11. This results in an approximately exponential distribution of the bearing failure data. In Eq. (8), Lββββ is the characteristic life or the life at which 63.2 percent of the bearings fail.

It was assumed that for each of the two bearing types described in Table 1 there are three virtual bins containing components from which the bearing was assembled. As in a realistic manufacturing process, each of the component parts of the respective bearings are grouped in separate bins. Each bin contains either 1000 inner rings, outer rings or ball sets. Each component part and ball set is assigned an order number (1, 2, 3, �.1000) corresponding to its life correlated to the respective Weibull plots for the components shown in Fig. 1.

959080

60

40

20

10

6432

Bea

ring

s o

r co

mp

one

nts

faile

d, p

erce

nt

1 2 4 6 10 20Life, hr

40 100 200 1000�103

Bearing

Inner race

Outer race

Balls

Angular-contact ball bearing

959080

60

40

20

10

6432

1 2 4 6 10 20Life, hr

40 100 200 1000�103

Bearing

Inner race

Outer race

Balls

Deep-groove ball bearing

(a)

(b)

Fig. 1.—Weibull plots for lives of deep-groove and angular-contact ball bearings and their respective components. Bore size, 50 mm: maximum Hertz stress, 1.55 GPa (225 ksi); speed, 10 000 rpm. (a) Deep-groove ball bearing; radial load, 950N (214 lb). (b) Angular-contact ball bearing; thrust load, 2800N (630 lb).

NASA/TM�2003-212186 6

Using Monte Carlo techniques, bearings were randomly assembled from the three virtual part bins for each bearing type. A 3-by-n matrix was randomly generated using the spreadsheet RAND function where n was the desired number of bearings to be assembled. The life of each individual bearing based upon the weak link theory was determined as being the lowest life of the randomly selected component of that bearing. Values of n were arbitrarily selected to be 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 35, 40, 45, 50, 100, 200, and 1000. All bearings in each set failed; there were no suspensions or censoring. Using the method of Johnson (11), the resulting lives of each group of n bearings were plotted on Weibull plots. A straight line was fitted to the data points using the method of least squares. The Weibull slope and the L10 and L50 lives were determined from the resultant line. A representative Weibull plot comprising 30 bearings is shown in Fig. 2. For each value of n, the procedure was repeated 10 times to estimate variation between trials and to determine the maximum and minimum values for the series of 10 trials.

RESULTS AND DISCUSSION

Endurance Life Variation

A. Palmgren in the 1945 edition of his book (10) presents an excellent discussion of bearing life. If a bearing is

properly designed, lubricated, operated, and maintained, its service life is limited by rolling-element fatigue. In the classical sense, a crack initiates in the subsurface zone of maximum shearing stresses below the running track of one of its elements and propagates into a pit or spall that is initially limited in area to that of the Hertzian contact area and to the depth of the maximum shearing stresses (1).

The time it takes for this spall to occur is the measure of the bearing life. According to Palmgren (10), bearing life is measured in number of revolutions of the bearing or in the number of hours of bearing operation at a given speed. Palmgren (10) states that �it is necessary to weigh, in a suitable manner, the contrary requirements of reliable service and low cost. Therefore, it has been decided to define the �estimated life� as that number of bearing revolutions or that number of working hours at a certain speed of rotation, which will be reached by 90 percent of all bearings.� Palmgren first proposed this definition of bearing life in 1924 (12,13). Today, this is the universally accepted definition for most bearing life calculations. As far as we can determine, it is the first probabilistic approach to life prediction of machine elements.

For our study bearings were randomly assembled from the three virtual part bins for each of the two bearing types by Monte Carlo (random) number generation. The life of each individual bearing based upon the weak link theory was determined as being the life of the lowest lived randomly selected component of that bearing. Using the method of Johnson (11), the resulting lives of each group of n bearings were plotted on Weibull plots and the Weibull slope and lives were determined (Fig. 2). For each value of n, the procedure was repeated 10 times to determine maximum values, minimum values, and the degree of variation between trials. The results are summarized in Table 2.

Table 2 contains the maximum and minimum values of the virtual L10 bearing lives of 10 trials of each group of n bearings with corresponding values for Weibull slope e and L50 bearing life. The calculated value was determined using the Palmgren-Lundberg equation and a life factor for lubrication�all other life factors were set to unity. The maximum and minimum values of the L10 lives as a percent of the calculated L10 life for each group of n bearings was determined as follows

9599

9080

60

40

20

10

6432S

tatis

tical

per

cent

of

bea

ring

s fa

iled

0.4 1 2Life, hr

4 6 10 20 40 80 200�103

Monte Carlo

Calculated (from Fig. 1(a))

Fig. 2.—Representative Weibull plot for Monte Carlo bearing tests of 50-mm bore deep-groove ball bearings compared to Weibull plot of calculated lives. Bore size, 50 mm: maximum Hertz stress, 1.55 GPa (225 ksi); speed, 10 000 rpm; L10 life, 3938 hr; L50 life, 37 087 hr; Weibull slope e, 0.84; failure index, 30 out of 30.

NASA/TM�2003-212186 7

Maximum Variation from Calculated L10 Life = percent 100

×

−

10

1010

L

LL

Calculated

CalculatedMaximum (9a)

Minimum Variation from Calculated L10 Life = percent 100

×

−

10

1010

L

LL

Calculated

CalculatedMinimum (9b)

Table 2. Summary of minimum and maximum life values from Monte Carlo simulation of assembly and testing of 340 sets of 50-mm bore, deep-groove, and angular-contact ball bearings

Maximum and minimum values of bearing life, hrs, and Weibull slope, e

Variation from calculated L10 life, percent

Maximum Corresponding Minimum Corresponding

Number of

bearings in a set L10 L50 e L10 L50 e

L10 lives below that

calculated, percent

Maximum Minimum

Deep-groove ball bearings 2 38709 66136 3.52 375 13478 0.53 40 461 �95 3 26767 65208 2.12 272 16186 0.46 70 286 �96 4 19331 78650 1.34 279 8500 0.55 60 180 �96 5 14091 63944 1.25 2619 15980 1.04 70 104 �59 6 23138 46278 2.72 270 14610 0.47 40 235 �96 10 10419 46713 1.26 1927 13845 0.96 80 51 �72 20 13538 55996 1.33 1176 13118 0.78 70 96 �82 22 12750 47314 1.44 1858 30048 0.68 60 84 �73 24 8526 41966 1.18 2886 27613 0.83 60 23 �58 26 10224 42022 1.33 4961 36330 0.95 40 48 �28 28 17635 54305 1.68 3946 29727 0.93 70 155 �43 30 10362 43021 1.32 3940 37090 0.84 50 50 �43 40 10318 36939 1.48 6769 34633 1.15 20 49 �11 50 9469 41669 1.27 3956 29150 0.94 50 37 �43 100 9936 51583 1.14 5033 32938 1.00 50 44 �27 200 9217 42421 1.23 6663 37852 1.08 30 33 �4 1000 7754 40660 1.14 6390 34511 1.18 10 12 �11

Angular-contact ball bearings 2 3696 4269 13.08 39 2906 0.44 60 283 �95 3 4612 17330 1.42 96 5026 0.48 60 378 �90 4 6194 15929 2.00 293 3674 0.75 40 542 �70 5 4203 7164 3.53 279 6267 0.61 40 336 �71 6 4404 10753 2.11 565 5013 0.86 40 357 �41 10 2179 8112 1.43 105 2648 0.58 40 126 �89 20 1612 6005 1.43 341 3908 0.77 70 67 �65 22 1756 6799 1.39 241 4205 0.66 80 82 �75 24 1903 8741 1.24 465 3746 0.90 40 97 �52 26 1431 5526 1.39 463 4551 0.84 70 48 �52 28 1560 5860 1.42 579 5246 0.86 70 62 �40 30 1423 5651 1.37 609 4289 0.97 50 48 �37 40 1701 7281 1.30 476 4835 0.81 80 76 �51 50 1406 7275 1.15 708 4785 0.99 40 46 �27 100 1471 6889 1.22 751 4708 1.03 40 53 �27 200 1179 5597 1.21 852 5210 1.04 60 22 �12 1000 1101 5429 1.18 917 5067 1.10 30 14 �5

NASA/TM�2003-212186 8

The life results and variations are summarized in Table 2. The variations were plotted on semi-log paper and are shown in Figs. 3(a) and 3(b) for the deep-groove and angular-contact ball bearings, respectively. Best-fit curves obtained using the linear regression package of a commercial spreadsheet were fitted through the points for the minimum and maximum values shown in each of the plots. The results for both sets of bearings were nearly identical. From these curve fits and Eqs. (9a) and (9b), the life equations from each set of the respective curves were as follows

Maximum L10 Life = Calculated L10 Life (1 + 6n�0.6)

(10a) Minimum L10 Life = Calculated L10 Life (1 � 1.5n�0.33) where n > 3 (10b) Minimum L10 Life → 0 where n ≤ 3

These life equations hold for any bearing geometry with a known calculated L10 life.

Relating these results to experimental research, the resultant L10 life should fall between these values if the true life is no different from the analytically predicted (calculated) life. If, however, the L10 life is greater than the Maximum Variation L10 Life, then the true life is probably greater than that calculated. If the L10 life is less than the Minimum Variation L10 Life, it must be reasonably concluded that the true L10 life is probably less than that calculated.

The data were studied to determine if the number of bearings tested affects whether the resultant L10 life will be less or more than that calculated. Out of the 170 trials comprising 15700 angular-contact ball bearings, the resultant L10 life was less than that calculated 54 percent of the time. For 170 trials comprising 15700 deep-groove ball bearings, the resultant L10 life was less than that calculated 51 percent of the time. The variation was random and independent of the number of bearings tested.

From this Weibull-based Monte Carlo study, for the thrust-loaded angular-contact ball bearings it was found that 45.4 percent of the failures occur on the inner race,

45.2 percent occur on the balls, and 9.4 percent on the outer race. For the radially loaded deep-groove ball bearings, 70.1 percent of the failures occur on the inner race, 15.1 percent on the balls, and 14.8 percent on the outer race. The failure locations for radially loaded cylindrical roller bearings are expected to be similar to those of the deep-groove ball bearing.

Similar failure trends in the percentage of individual components failed with respect to total system can be derived from the Lundberg-Palmgren model for system failure (5). The percentage of the bearing failures that are due to failure of the inner race can be derived using Eq. (5) and expressed as

100

1

1100 xeex

efailuresrace-innerpercent

or

ir

re

irir

sys

++

=

=

10

10

10

1010

10

L

L

L

LL

L (11)

Vari

atio

n fr

om

cal

cula

ted li

fe, p

erce

nt600

400

–200

0

200

1 2 4 6 10 20Number of bearings failed, n

40 100 200 400 1000

Calculated life

Min variation

Max variation

Deep-groove ball bearings–Monte Carlo

Fig. 3.—Maximum and minimum variation of L10 lives as percent of calculated L10 for each group of n bearings compared with 90% confidence limits based on Weibull slope and number of bearings failed n. (a) 50-mm bore deep-groove ball bearing. (b) 50-mm bore angular-contact ball bearing. (c) 90% confidence limits.

600

400

–200

0

200


40 100 200 400 1000

Calculated life

Min variation

Max variation

Angular-contact ball bearings–Monte Carlo

600

800

1000

1200

400

–200

0

200


40 100 200 400 1000

Calculated life

Min variation(from Fig 3(a) and 3(b))

Max variation(from Fig 3(a) and 3(b))

90% Confidence limit(from Johnson (11)) for

Weibull slope, e, of

(a)

(b)

(c)

11.52

Upper limit

Lower limit

NASA/TM�2003-212186 9

Table 3. Comparison of bearing component failure distributions based upon a Weibull-based Monte Carlo method and calculated from Lundberg-Palmgren system life equation for deep-groove and angular-contact ball bearings

Percent Failure Bearing Type Component Weibull-based Monte

Carlo result Calculated from Lundberg-

Palmgren Eq. (5) Inner Race 70.1 69.9 Rolling Element 14.8 15.0

Deep-Groove Ball Bearing Outer Race 15.1 15.0

Inner Race 45.4 45.1 Rolling Element 45.2 45.1

Angular Contact Ball Bearing

Outer Race 9.4 9.7 where L10 sys is the life at which 10 percent of the assembled systems have failed, L10 ir is the life at which 10 percent of the inner rings have failed, L10 or is the life at which 10 percent of the outer rings have failed, and e is the Weibull slope. A derivation of Eq. (11) is provided in the appendix. The percentage of rolling elements failed or the percentage of outer races failed can be expressed similarly (see the appendix). Table 3 is a summary of the percentage of inner race, rolling element, and outer race failures obtained from the Weibull-based Monte Carlo method and Eq. (11). There is excellent agreement between these techniques.

Assuming that the calculated L10 life is exact and not subject to unknown manufacturing or operating variables, there is an even (50 %) probability that the resultant L10 life for any randomly selected bearing test group will be less than that calculated. As the number of bearings in a test group increases, the resultant life approaches the predicted life but with half the lives still being less than that calculated.

Confidence Limits

The results of the above analysis were compared by us to that of L. Johnson (11) who determined the 90-percent

confidence limits as a function of Weibull slopes of 1, 1.5, and 2 and the number of failed bearings n in a set. These confidence limits for the L10 life are plotted in Fig. 3(c). The confidence limits designate that the true population of the L10 lives will fall between the upper and lower values in 90 percent of all possible cases or tests (11). An approximate curve fit of the Johnson analysis results in the following approximate equations for the upper and lower values of the L10 life 90-percent confidence limits

For Weibull slope = 1

Upper limit: L10up = calculated L10 + 26.5 calculated L10 /n0.8 (12a)

Lower limit: L10L = calculated L10 � calculated L10(1.2 � 0.17lnn) (12b) L10L → 0 where n ≤ 3 (12c)

For Weibull slope = 1.5


Lower limit: L10L = calculated L10 � calculated L10(1 � 0.15lnn) (13b) L10L = calculated L10 where n ≥ 786 (13c)

For Weibull slope = 2


Lower limit: L10L = calculated L10 � calculated L10(0.87 � 0.13lnn) (14b) L10L = calculated L10 where n ≥≥≥≥ 800 (14c)

NASA/TM�2003-212186 10

The results from Figs. 3(a) and 3(b) were superimposed in Fig. 3(c). What is significant is that the minimum variation values of the L10 life coincide with the lower 90-percent confidence limits for a Weibull slope of 1. However, there are small differences in life between the lower life limits for Weibull slopes of 1, 1.5, and 2. The 90-percent confidence limits of L10 life for a Weibull slope of 1.5 best correlated with the Monte Carlo results for the upper and lower life variations as a function of the number of bearings tested to failure and is independent of whether the bearing is thrust or radially loaded.

Maximum Likelihood Estimators (MLE)

The curve fits of the upper and lower bounds of the Monte Carlo generated bearing lives (Eqs. (10a) and (10b))

and the tabulated 90% confidence limits for a Weibull slope of 1.5 from Johnson (11) (Eqs. (13a) and (13b)) are in excellent agreement. These curves are expressed in easy to apply algebraic equation defining bearing life variation. As will be discussed, bearing data available in the open literature (7,8) reasonably fall between these easily established limits. Another method for calculating these upper and lower bounds is maximum likelihood estimators (MLE).

MLE can be obtained from the methods of Cohen (14), Harter and Moore (15), and McCool (16), from which confidence limits can be established (17). Figure 4 includes the upper and lower bounds established from the Weibull-based Monte Carlo technique (Eqs. (10a) and (10b)), the 90%-confidence limits based upon Johnson (Eqs. (14a) and (14b)) and the confidence limits based upon MLE. In general, there is good agreement among these techniques.

The confidence levels based upon the MLE are more complicated to calculate than either Eqs. (10a) and (10b) or (14a) and (14b). The MLE is an iterative process. It can be sensitive to the choice of starting values, and the calculation is usually non-trivial, tending to require the use of computational software. It should be noted that the maximum likelihood limits can be biased for small sample sizes. Additionally, the information available in the open literature (17) limits the assignment of confidence limits and the application of these techniques to several narrowly defined cases of limited engineering application.

Of the 340 bearing population studied by us and bounded by both the Monte Carlo results (Eqs. (10a) and (10b)) and the 90% confidence limit curve fits of Johnson (Eqs. (13a) and (13b)), the confidence limits based upon MLE can be determined for only 8 cases without extensive additional Monte Carlo simulations. It is worth noting that an engineering approach to confidence intervals has been proposed by Houpert (18), and is based upon a linear regression curve fitting technique. The technique is more complex than the algebraic equations (Eqs. (10a), (10b), (13a), and (13b)) presented by us. A comparison between our technique and that of Houpert (18) is beyond the scope of this paper.

Vari

atio

n fr

om

cal

cula

ted li

fe, p

erce

nt

Fig. 4.—Comparison of 90% confidence limits from maximum likelihood estimates (MLE) (14,17) of maximum and minimum variation of L10 lives as percent of calculated L10 to that of Johnson (11) and Monte Carlo method for 50-mm bore deep-groove ball bearing.

200

250

150

–100

–50

0

100

50

0 10 20Number of bearings failed, n

30 40

Calculated life

Monte Carlo (from Fig 3(c))Johnson, e = 1.5 (from Fig 3(c))MLE

Upper limit

Lower limit

NASA/TM�2003-212186 11

Weibull Slope Variation Lundberg and Palmgren (5) assumed the value of the

Weibull slope e in Eq. (1) to be 1.11. This value was necessary in their analysis because it approximated that value exhibited by their experimental data and it made the end result of their life prediction analysis correlate with their bearing life database at that time. Experience has shown that most rolling bearing life data exhibit Weibull slopes between 1 and 2. For the analysis performed by us, we too assumed a Weibull slope of 1.11 for all of the components for each bearing. This should theoretically result in a bearing Weibull slope of 1.11 as shown in Fig. 1 for the deep-groove and angular-contact ball bearings.

Johnson (11) analyzed the probable variation of the Weibull slope as a function of the number of bearings tested to failure. Based on the Johnson analysis (11), in 90 percent of all possible cases the resultant Weibull slope will be within the limits shown in Fig. 4 based upon a Weibull slope of 1.11. Based on Johnson (11), the approximate relation for the number of bearings failed n and the limits of the value of Weibull slope e equal to 1.11 are as follows:

Maximum Weibull slope = 1.11 + 1.31 n�0.5 (15a) Minimum Weibull slope = 1.11 � 1.31 n�0.5 (15b)

The results of the extremes in the Weibull slopes for

each group of the ten bearing trials of n bearings are compared with the Johnson analysis in Fig. 5(a). Note that the Weibull slopes for the data summarized in Table 2 for the maximum and minimum bearing lives are not necessarily the same as the maximum and minimum values of the Weibull slopes for each of trials of n bearings. For the data reported in Table 2 the relation between the number of bearings tested and the limits of the Weibull slope are as follows:

Maximum Weibull slope = 1.2 + 5(lnn)�3 (16a) Minimum Weibull slope = 1.11 � 0.95n�0.33 (16b) Where the number of bearings failed is 10 or greater, there is a reasonably good correlation between the limits of the slopes

generated from the Johnson analysis (11) and those from our Monte Carlo bearing tests. Where the number of failed bearings is below 10, there are differences between the extremes in Weibull slope between the Monte Carlo bearing tests and those of Johnson, especially at the upper limits for the Weibull slopes.

Comparison With Bearing Data

Harris (7,8) analyzed 62 rolling-element bearing endurance sets. These data were obtained from four bearing

manufacturers, two helicopter manufacturers, three aircraft engine manufacturers, and U.S. Government agency-sponsored technical reports. The data sets comprised deep-groove radial ball bearings, angular-contact ball bearings, and cylindrical roller bearings for a total of 7935 bearings. Of these, 5321 bearings comprised one sample size for a single cylindrical roller bearing leaving 2614 bearings distributed among the remaining bearing types and sizes. Among the 62 rolling-element bearing endurance sets, 11 had one or no failure and could not be used for our analysis. These data are summarized in Table 4. A discussion of the Harris data can be found in Refs. (6) and (9).

Wei

bul

l slo

pe,

eW

eib

ull s

lop

e, e

Fig. 5.—Variation of Weibull slope e compared to predicted 90% probable error. (a) Extremes of Weibull slope from Monte Carlo testing for each group of ten bearing trials of n bearings. (b) Weibull slopes from 51 sets of ball and roller bearing data from ref. (7,8).

1.6

2.0

2.4

2.8

1.2

0.0

0.4

0.8


40 100 200 400 1000

e = 1.11

Lower 90% probable error predicted from ref. (11)

Upper 90% probable error predicted from ref. (11)

(b)

3.633.48

1.6

2.0

2.4

2.8

1.2

0.0

0.4

0.8


40 100 200 400 1000

e = 1.11

Lower 90% probable error predicted from ref. (11)

Upper 90% probable error predicted from ref. (11)

(a)

13.08 3.53

Deep-grove ball bearingAngular-contact ball bearingCylindrical roller bearing

NASA/TM�2003-212186 12

Table 4. Summary of rolling-element bearing life data for three bearing types

(data from Ref. (7)) Bearing set

identification numbera

Maximum Hertz

stress, GPa (ksi)

Ratio of actual L10

life to calculatedb

Weibull slope,

e

Failure indexc

L10 life variation

from calculated,

percentd

Steel and processing

Deep-groove ball bearing; load-life exponent, p = 3 6 3.13 (454) 3.42 2.29 11/18 242 VARe

AISI M�50 18 4.03 (584) 8.77 2.22 3/6 777 CVDf

AISI 52100 28 3.41 (495) 4.88 2.29 7/30 388 VIM-VARg

AISI M�50 29 3.41 (495) 9.88 1.06 3/28 888 VIM-VAR

AISI M�50 30 3.55 (515) 4.67 0.72 6/37 367 VIM-VAR

AISI M�50 40 3.41 (495) 0.89 0.51 11/40 �11 CVD

AISI 52100 41 3.41 (495) 6.85 0.70 2/41 585 CVD

AISI 52100 42 3.41 (495) 3.80 2.65 23/37 280 CVD

AISI 52100 43 3.88 (563) 0.89 1.28 7/11 �11 CVD

AISI 52100 44 3.41 (495) 1.51 0.89 7/37 51 CVD

AISI 52100 46 3.88 (563) 2.99 2.22 22/40 199 CVD

AISI 52100 47 3.41 (495) 2.53 0.65 4/33 153 CVD

AISI 52100 48 3.41 (495) 0.52 3.48 33/40 �48 VIM-VAR

AISI M�50 49 3.41 (495) 1.50 0.68 6/40 50 VIM-VAR

M�50NiL 50 3.41 (495) 4.98 1.23 5/40 398 VIM-VAR

M�50NiL 51 3.1 (539) 3.16 1.33 3/28 216 CVD

AISI 52100 53 3.12 (452) 1.53 0.65 9/67 53 CVD

AISI 52100 54 3.72 (539) 2.11 0.93 21/60 111 CVD

AISI 52100 55 3.72 (539) 1.44 0.95 57/57 44 CVD

AISI 52100 56 3.72 (539) 5.21 0.72 8/30 421 CVD

AISI 52100 57 3.72 (539) 2.32 0.70 12/30 132 CVD

AISI 52100 58 3.72 (539) 4.85 0.69 8/29 385 CVD

AISI 8620 59 3.72 (539) 5.88 1.20 12/29 488 CVD

AISI 8620 60 3.72 (539) 0.38 2.75 29/29 �62 CVD

AISI 52100 61 2.97 (431) 3.27 1.36 43/103 227 CVD

AISI 52100

NASA/TM�2003-212186 13

Table 4. Summary of rolling-element bearing life data for three bearing types (data from Ref. (7)) (continued)

Bearing set identification

numbera

Maximum Hertz

stress, GPa (ksi)

Ratio of actual L10

life to calculatedb

Weibull slope,

e

Failure indexc

L10 life variation

from calculated,

percentd


62 2.97 (431) 1.45 1.21 23/79 45 CVD AISI 52100

Angular-contact ball bearings; load-life exponent, p = 3 1 3.65 (530) 7.63 0.14 2/12 663 VIM-VAR

M�50NiL

2 1.69 (245) 96 1.15 3/199 9500 VIM-VAR AISI M�50

4 2.01 (292) 3.26 0.84 5/17 226 VAR AISI M�50

8 2.07 (300) 4.46 1.14 2/10 346 VAR AISI M�50

9 2.34 (339) 1.11 0.84 5/10 11 VAR AISI M�50

10 2.34 (339) 1.04 0.69 4/10 4 VAR AISI M�50

12 2.70 (391) 3.37 1.91 5/8 337 VAR AISI M�50

32 2.23 (324) 6.33 1.08 3/30 533 VIM-VAR AISI M�50

33 1.97 (286) 1.11 0.81 3/20 11 VIM-VAR AISI M�50

34 1.97 (286) 2.26 1.10 2/20 126 VIM-VAR AISI M�50

35 1.25 (181) 2.08 0.80 2/64 108 VIM-VAR AISI M�50

37 1.28 (185) 3.46 0.95 7/362 246 VIM-VAR AISI M�50

38 1.28 (185) 101.62 0.67 2/634 10062 VIM-VAR AISI M�50

39 1.28 (185) 12.85 0.20 2/33 1185 VIM-VAR AISI M�50

Cylindrical roller bearings; load-life exponent, p = 10/3 3 1.22 (177) 36 1.00 13/5321 3500 VIM-VAR

AISI M�50 5 2.15 (312) 90 0.21 2/19 8900 VAR

AISI M�50 14 2.45 (356) 16 3.63 6/6 1500 CVD

AISI 52100 15 2.45 (356) 2165 2.43 6/6 2065 VAR

AISI M�50 16 2.39 (346) 1.57 0.74 4/7 57 CVD

AISI 52100 17 2.39 (346) 6.49 0.48 2/8 549 VAR

AISI M�50 20 2.23 (324) 6.94 1.08 3/6 594 CVD

AISI 52100 22 2.34 (339) 4.78 1.90 3/6 378 CVD

AISI 52100 24 2.29 (332) 10.5 0.75 4/6 905 CVD

AISI 52100 26 2.35 (341) 11.23 1.23 4/6 1023 CVD

AISI 52100

NASA/TM�2003-212186 14

Table 4. Summary of rolling-element bearing life data for three bearing types (data from Ref. (7)) (concluded)

Bearing set identification

numbera

Maximum Hertz

stress, GPa (ksi)

Ratio of actual L10

life to calculatedb

Weibull slope,

e

Failure indexc

L10 life variation

from calculated,

percentd


27 2.35 (341) 7.63 0.67 4/6 663 VAR AISI M�50

aRefers to bearing sets as identified in Refs. (7) and (8) bLife calculation based on Lundberg-Palmgren equations with STLE life factors cNumber of bearings failed out of number of bearings in a set dRefer to Eqs. (2) and (9) eVacuum arc remelting fCarbon vacuum degassing gVacuum induction melting, vacuum arc remelting

Vari

atio

n fr

om

cal

cula

ted

life

, per

cent

Fig. 6.—Variation between actual and calculated L10 bearing lives for 51 sets of deep-groove and angular-contact ball bearings and cylindrical roller bearings from ref. (7,8) compared to Monte Carlo variations and 90% confidence limit.

600

800

1000

1200

400

–400

–200

0

200


40 100 200 400 1000

Min variation (from Fig 3)

Max variation (from Fig 3)

Deep-grove ball bearingAngular-contact ball bearingCylindrical roller bearing

Upper 90% confidencelimit for e = 1 variation(from Fig 3(c))

There is insufficient technical information reported in Refs. (7) and (8) regarding bearing geometry, operating conditions, fit-ups, and material properties as well as individual data points and Weibull plots to independently evaluate the life results reported. However limited, these results are the best compilation of bearing life data in the open literature.

The data of Table 4 are plotted in Fig. 6 using the number of bearings failed rather than the number of bearings in a set. These data are compared to the maximum and minimum life variations of Fig. 3 and the 90-percent confidence upper limit for a Weibull slope of 1 from Fig. 3(c) and Eq. (13a). Figure 6 consists of 51 sets of bearing data. These data suggest a greater variation between calculated and resultant L10 lives than that of the Monte Carlo virtual bearing tests of Figs. 3(a) and 3(b).

Of these data, 39 percent fall between the maximum and minimum life variations suggesting that the statistical variations of these lives are within that predicted. Four bearing sets representing 8 percent of the bearing sets had lives less than that predicted. Thirty bearing sets or 59 percent of the bearing sets exceeded the maximum life variation of this Monte Carlo study. Eight of these bearing sets or 16 percent exceeded the 90-percent confidence upper limit of Johnson. However, only one bearing set representing 2 percent of the bearing sets fall below the lower life limit. Therefore, it can be reasonably concluded that 98 percent of the bearing sets have acceptable life results using the Lundberg-Palmgren equations with the life adjustment factors from Ref. (1) to predict bearing life.

NASA/TM�2003-212186 15

Va

riat

ion

fro

m c

alcu

late

d li

fe, p

erce

nt

Fig. 7.—Variation between actual and calculated L10 bearing lives for deep-groove ball bearings as a function of load-life exponent p. (a) Load-life exponent p = 3 (from Fig. 6). (b) Load-life exponent p = 3.5 and 4.

600

1000

1200

400

–400

–200

0

200


40 100 200 400 1000



(b)

800

Load-lifeexponent,

p33.54

600

800

1000

1200

400

–400

–200

0

200


40 100 200 400 1000




(a)

Vari

atio

n fr

om

cal

cula

ted

life

, per

cent

Fig. 8.—Variation between actual and calculated L10 bearing lives for angular-contact ball bearings as a function of load-life exponent p. (a) Load-life exponent p = 3 (from Fig. 6). (b) Load-life exponent p = 3.5 and 4.

600

800

400

–400

–200

0

200


40 100 200 400 1000

600

800

1000

1200

400

–400

–200

0

200


40 100 200 400 1000




(a)

(b)Min variation (from Fig 3)


Load-lifeexponent,

p33.54

Figure 7(a) consists only of the deep-groove ball bearings from Fig. 6. There are 26 deep-groove ball bearing sets comprising 51 percent of all bearing sets. Of these 11 bearings sets or 42 percent are between the maximum and minimum life values. One bearing set representing 4 percent of the deep-groove ball bearings falls below the lower life value. Fourteen or 54 percent of the bearing sets exceeded the maximum life value. Three of these bearing sets or 12 percent exceeded the 90-percent confidence upper limit. For the deep-groove bearing sets, it can also be concluded that by using the Lundberg-Palmgren equations with life adjustment factors to predict bearing life, 96 percent of the bearing sets have acceptable life results.

Figure 8(a) consists only of the angular-contact ball bearings from Fig. 6. There are 14 angular-contact ball bearing sets comprising 27 percent of all bearing sets. All of the bearing sets equaled or exceeded the predicted life value. Of these 8 bearings sets or 57 percent are between the predicted and maximum life variation. Six or 43 percent of the bearing sets exceeded the maximum life variation. Three of these bearing sets or 14 percent exceed the 90-percent confidence upper limit. For the angular-contact ball bearing sets, it can also be concluded that by using the Lundberg-Palmgren equations with life adjustment factors used to calculate bearing life, 100 percent of the bearing sets have acceptable life results.

Based upon the work of Zaretsky, Poplawski, and Peters (6, 19, 20) it can be concluded that the life of these bearings are under-predicted. They suggest that for ball bearings the load-life exponent p from Eq. (2) should be 4 instead of 3 used by Lundberg and Palmgren (5) and reflected in the data of Figs. 7(a) and 8(a). The life data for the ball bearings of Figs. 7(a) and 8(a) were recalculated using load-life exponent values p of 3.5 and 4. The results are shown and compared in Figs. 7(b) and 8(b) for the deep-groove and angular-contact bearings, respectively. A load-life exponent of 4 best reflects the variation in the ratio of the actual life to the predicted life and p = 3 is conservative commensurate with good engineering practice.

NASA/TM�2003-212186 16

Figure 9(a) comprises only the cylindrical roller bearings from Fig. 6. There are 11 cylindrical roller bearing sets comprising 22 percent of all bearing sets. Of these, one bearing set or one percent is between the maximum and minimum life variation. No bearing set falls below the lower life variation. Ten or 91 percent of the bearing sets exceeded the maximum life variation and 6 of these bearing sets or 55 percent exceeded the 90-percent confidence upper limit. For the cylindrical roller bearing sets, it can also be concluded that by using the Lundberg-Palmgren equations with life adjustment factors to predict bearing life, 100 percent of the bearing sets have acceptable life results.

Referring again to Eq. (2), the value for the load-life exponent p for cylindrical roller bearings as used by Lundberg and Palmgren is 10/3 or 3.333. Poplawski, Peters and Zaretsky (19,20) stated that based upon their experience and analysis, the load-life exponent p of 10/3 is incorrect and will under predict roller bearing life. It was their recommendation that this value be revised to 4 with consideration given to increasing it to 5. Based their recommendation, the data for the cylindrical roller bearings from Fig. 9(a) were recalculated by us using values of p equal to 4, 4.5, and 5. These results are shown in Fig. 9(b). A load-life exponent of p = 5 best reflects the cylindrical bearing life results that are reported. However, by using a value of p = 4 a more conservative life prediction results that may be more commensurate with good engineering practice.

Table 5 shows the effect on bearing life using different values of the load-life exponent p for ball and roller bearings at three load conditions. The table was normalized to the light load condition where Peq = 0.05 CD. What is apparent from this table and

Table 5. Effect of load-life exponent, p, on bearing life prediction

Relative life Load-life exponent, p

Ball bearings Roller bearings

Bearing load Peq,

percent dynamic load capacity CD 3a 3.5 4 10/3b 4 4.5 5

Light load, 5 1 4.47 20 1 7.4 33 148 Normal load, 10 0.13 0.40 1.25 0.10 0.46 1.46 4.61 Heavy load, 20 0.02 0.03 0.08 0.01 0.03 0.06 0.14

Life factor Light load, 5 20c 4.5 1 148d 20 4.5 1 Normal load, 10 10 3.2 1 46 10 3.2 1 Heavy load, 20 5 2.2 1 15 5 2.2 1 aNormalized to load-life exponent p = 3 and light load bNormalized to load-life exponent p = 10/3 and light load cNormalized to load-life exponent p = 4 dNormalized to load-life exponent p = 5

Vari

atio

n fr

om

cal

cula

ted

life

, per

cent

Fig. 9.—Variation between actual and calculated L10 bearing lives for cylindrical roller bearings as a function of load-life exponent p. (a) Load-life exponent p = 10/3 (from Fig. 6). (b) Load-life exponent p = 4, 4.5, and 5.

600

800

1000

1200

400

–400

–200

0

200


40 100 200 400 1000

600

800

1000

1200

400

–400

–200

0

200


40 100 200 400 1000




(a)

(b)Min variation (from Fig 3)


Load-lifeexponent,

p10/344.55

NASA/TM�2003-212186 17

the bearing data is that there is built into the Lundberg-Palmgren life calculations a very conservative safety factor depending on the load. For lightly loaded ball bearings, this factor could be as high as 20 and for heavy loaded ball bearings as much as 4. For roller bearings, the factor for lightly loaded bearings is 149 and heavy loaded bearings as much as 14.

Referring to the bearing data of Table 4 and Fig. 6, only 4 sets of data are less than that predicted and only two significantly. This results in 92 percent of the bearing sets equaling and/or exceeding the currently predicted life. Should the load-life exponent p be changed to its correct value and more correctly predict bearing life, there is a probability that 50 percent of the bearings in small subsets would not reach their predicted life in actual application. From a design engineering approach, it would be prudent engineering practice to maintain the load-life exponent p = 3 for ball bearings. However, we suggest that the load-life exponent p = 10/3 for roller bearings be changed to p = 4. This would result in each bearing type having similar life factors for purposes of design as shown in Table 5.

The Weibull slopes for each of the bearings are plotted in Fig. 5(b). While most of the scatter falls within the predicted range, approximately 25 percent of the Weibull slopes for the bearing data fall outside that predicted. These results correlate with the Weibull slopes obtained with the Monte Carlo bearing virtual data. It can be concluded that larger deviations in bearing Weibull slope can occur than predicted by Johnson�s method based on a Weibull slope of 1.11, even with bearing sets of n equal 30.

Material and Processing Effects

The life calculations for the data of Table 4 and Fig. 6 have material and steel processing life factors incorporated in them from Ref. (1). Table 6 breaks down and summarizes these life factors for each of the materials listed in Table 4. The data of Table 4 and Fig. 6 are broken down and plotted in part (a) of Figs. 10 through 13 based on material and steel processing variables. These data were adjusted for a load-life exponent p of 4 for ball bearings and 5 for roller bearings and are shown in part (b) of Figs. 10 through 13. The adjusted life results correlated with those of the Monte Carlo tests shown in Fig. 3. Based upon these material and processing life factors and load-life exponents, each bearing data set appears consistent with the other. However, for AISI 8620 steel, these data suggest that the material factor should be increased from 1.5 to 2.

Table 6. Life factors for bearing steels and processing (from Ref. (1))

Life factor Material and process Material Process Resultant

CVDa AISI 52100

3 1.5 4.5

CVDa AISI 8620

1.5 1.5 2.25

VARb AISI M�50

2 3 6

VIM-VARc AISI M�50

2 6 12

VIM-VARc M�50NiL

4 6 24

aCarbon vacuum degassing bVacuum arc remelting cVacuum induction melting, vacuum arc remelting

NASA/TM�2003-212186 18

Vari

atio

n fr

om

cal

cula

ted

life

, per

cent

Fig. 10.—Effect of CVD AISI 52100 steel and load-life exponent on bearing life. (a) Load-life exponent p, 3 for ball bearings; 10/3 for cylindrical roller bearings (from Fig. 6). (b) Load-life exponent p, 4 for ball bearings; 5 for cylindrical roller bearings.

600

800

400

–400

–200

0

200


40 100 200 400 1000

600

800

1000

1200

400

–400

–200

0

200


40 100 200 400 1000



Deep-grove ball bearingAngular-contact ball bearingCylindrical roller bearings

p = 4 for ball bearings; p = 5 for roller bearings


(a)

(b)

p = 3 for ball bearings; p = 10/3 for roller bearings



Vari

atio

n fr

om

cal

cula

ted

life

, per

cent

Fig. 11.—Effect of VAR AISI M-50 steel and load-life exponent on bearing life. (a) Load-life exponent p, 3 for ball bearings; 10/3 for cylindrical roller bearings (from Fig. 6). (b) Load-life exponent p, 4 for ball bearings; 5 for cylindrical roller bearings.

600

800

1000

400

–400

–200

0

200


40 100 200 400 1000

600

800

1000

1200

400

–400

–200

0

200


40 100 200 400 1000





(a)

(b)





NASA/TM�2003-212186 19

Vari

atio

n fr

om

cal

cula

ted

life

, per

cent

Fig. 12.—Effect of VIM-VAR AISI M-50 steel and load- life exponent on bearing life. (a) Load-life exponent p, 3 for ball bearings; 10/3 for cylindrical roller bearings (from Fig. 6). (b) Load-life exponent p, 4 for ball bearings; 5 for cylindrical roller bearings.

600

800

400

–400

–200

0

200


40 100 200 400 1000

600

800

1000

1200

400

–400

–200

0

200


40 100 200 400 1000





(a)

(b)





Vari

atio

n fr

om

cal

cula

ted

life

, per

cent

Fig. 13.—Effect of VIM-VAR M-50Nil and CVD AISI 8620 steels and load-life exponent on bearing life. (a) Load-life exponent p, 3 for ball bearings (from Fig. 6). (b) Load-life exponent p, 4 for ball bearings.

600

800

400

–400

–200

0

200


40 100 200 400 1000

600

800

1000

1200

400

–400

–200

0

200


40 100 200 400 1000

(a)

(b)

Deep-groove ball bearingAngular-contact ball bearing

Open symbols designate VIM AISI M-50NilSolid symbols designate CVD AISI 8620

p = 4 for ball bearings

p = 3 for ball bearings





Comparing Life Results to Prediction

Rules can be implied from the results of this paper to compare and distinguish tests of identical bearings either from two or more sources or made from different manufacturing methods. The following rules are suggested to determine if the bearings are acceptable for their intended application or if there are significant differences between the two groups of bearings.

1. If the L10 lives of both bearing tests are between the Maximum and Minimum L10 life variations, there can

be no conclusion that there is a significant difference between the two sets of bearings regardless of the ratio of the L10 lives. The bearing sets are acceptable for their intended application (Fig. 14(a)).

2. If the L10 life of one set of bearings is greater than the maximum variation and the second set is less the minimum value, there exists a significant difference between the bearing sets. Only, one bearing set is acceptable for its intended application (Fig. 14(b)).

3. If the L10 lives of both sets of bearings exceed the maximum variation, the bearing life differences may or may not be significant and should be evaluated based upon calculation of confidence numbers according to the method of Johnson (11). Both sets of bearings are acceptable for their intended application (Fig. 14(c)).

NASA/TM�2003-212186 20

4. If the L10 lives of both sets of bearings are less than the minimum variation, the bearing life differences may or may not be significant. However, both sets of bearings are not acceptable for their intended application (Fig. 14(d)).

5. If the L10 life of one set of bearings exceed the maximum variation and the other set is between the maximum and minimum variation, the bearing life differences may or may not be significant and should be evaluated based upon calculation of confidence numbers according to the method of Johnson (11). Both sets of bearings are acceptable for their intended application (Fig. 14(e)).

6. If the L10 life of one set of bearings is less than the minimum variation and the other set is between the maximum and minimum variation, there exists a significant difference between the bearing sets. Only, one set of bearings is acceptable for its intended application (Fig. 14(f)).

Fig. 14.—Rules for comparing bearing life results to calculated life. (a) A and B bearing sets are acceptable. (b) A bearing set is acceptable. B bearing set not acceptable. (c) A and B bearing sets are acceptable. (d) A and B bearing sets are not acceptable. (e) A and B bearing sets are acceptable. (f) A bearing set is acceptable. B bearing set not acceptable.

400

–200

0

200

2 4 6 10 20Number of bearings failed, n

40(a)

Bearing set

Calculated life

AB


Vari

atio

n fr

om

cal

cula

ted

life

, per

cent

400

–200

0

200


40(b)

400

–200

0

200


40(c)


400

–200

0

200


40(d)

Vari

atio

n fr

om

cal

cula

ted

life

, per

cent

400

–200

0

200


40(e)

400

–200

0

200


40(f)

NASA/TM�2003-212186 21

SUMMARY OF RESULTS

Two types of rolling-element bearings representing radial loaded and thrust loaded bearings were used for this study. Three hundred forty (340) virtual bearing sets totaling 31400 bearings were randomly assembled from three virtual part bins for each of the bearing types by Monte Carlo (random) number generation. The life of each individual bearing based upon the weak link theory was determined as being the life of the lowest lived randomly selected component of that bearing. Using the method of Johnson (11), the individual bearing lives for each group of n bearings were plotted on Weibull plots and the L10 lives and Weibull slopes were determined. The Monte Carlo results were compared with endurance data from 51 bearing sets comprising 5321 bearings. The following results were obtained:

1. A simple algebraic relationship was established for the upper and lower bearing L10 life limits as function

of number of bearings failed for any bearing geometry. 2. Assuming an ideal and accurate bearing life prediction procedure, randomly assembled and selected

bearings from a carefully controlled large bearing population will result in a fifty percent (50 %) probability that the resultant bearing life will be less than that calculated regardless of the number of failed bearings making up the randomly selected group. However, the variation of the resultant life from that calculated will decrease as the number of failed bearings in the randomly selected group increases.

3. The maximum and minimum variation between the bearing resultant life and the calculated life was found to correlate with the 90-percent confidence limits for a Weibull slope of 1.5 and is independent of whether the bearing is thrust or radially loaded.

4. Recalculating the lives for the actual bearing data using a load-life exponent p of 4 for ball bearings and 5 for roller bearings results in a reasonable correlation between the maximum and minimum values of the Monte Carlo generated bearing lives and the actual bearing life data. For design purposes, the Lundberg-Palmgren life calculations incorporating a load-life exponent p of 3 for ball bearings should be retained. However, the load-life exponent p for roller bearings should be changed from 10/3 to 4.

5. The STLE life factors for bearing steel and processing in conjunction with the Lundberg-Palmgren life equation provide a reasonable accounting of differences in the material chemistry and processing. However, the material life factor for AISI 8620 should be changed from 1.5 to 2.

6. Maximum and minimum variations in Weibull slope from the Monte Carlo testing and bearing endurance data correlated with predicted values and was a function of the number of bearing failures. The greater the number of failures the less variation in the Weibull slope from that predicted.

7. There is excellent agreement between the percentage of individual components failed from Monte Carlo simulation and that predicted from the method of Johnson.

REFERENCES

1. Zaretsky, E.V., STLE Life Factors for Rolling Bearings, STLE SP�34, Society of Tribologists and Lubrication

Engineers, Park Ridge, IL, pp. 1�45, 1992, pp 233�298. 2. Weibull, W., �A Statistical Theory of the Strength of Materials,� Ingeniors Etanskaps Akademien-Handlinger,

no. 151, 1939. 3. Weibull, W., �The Phenomenon of Rupture of Solids,� Ingeniors Vetenskaps Akademien, no. 153, 1939. 4. Weibull, W., �A Statistical Distribution Function of Wide Applicability,� ASME Jour. of Applied Mechanics,

18, 3, 1951, pp. 293�297. 5. Lundberg, G., and Palmgren, A., �Dynamic Capacity of Rolling Bearings,� Acta Polytechnica, 1, 3, Stockholm,

1947. 6. Zaretsky, E.V., Poplawski, J.V., and Peters, S.M., �Comparison of Life Theories for Rolling-Element

Bearings,� STLE Tribology Trans., 39, 2, 1996, pp. 237�248, 501�503. 7. Harris, T.A., �Final Report-Establishment of a New Rolling Bearing Contact Life Calculation Method,� U.S.

Naval Air Warfare Center, Aircraft Division Trenton, Contract No. N68335�93�C�0111, 1995. 8. Harris, T.A., and McCool, J.I., �On the Accuracy of Rolling Bearing Fatigue Life Prediction,� ASME Jour. of

Tribology, 118, 2, 1996, pp. 297�310.

NASA/TM�2003-212186 22

9. Zaretsky, E.V., Poplawski, J.V., and Miller, C.R., �Rolling Bearing Life Prediction-Past, Present and Future,� Proc. International Tribology Conf., Nagasaki, 2000, 1, Japanese Society of Tribologist, Tokyo, Japan, 2001, pp. 101�107.

10. Palmgren, A., Ball and Roller Bearing Engineering, SKF Industries, Inc., Philadelphia, PA, 1945. 11. Johnson, L.G., The Statistical Treatment of Fatigue Experiments, Elsevier Publishing Co., Amsterdam, 1964. 12. Palmgren, A., �Die Lebensdauer von Kugellagern (The Service Life of Ball Bearings),� Zeitschrift des Vereines

Deutscher Ingenieure, 68, 1924, pp. 339�341. 13. Zaretsky, E.V., �A. Palmgren Revisited � A Basis for Bearing Life Prediction,� Lubrication Engineering, J.

STLE, 54, 2, Feb. 1998, pp. 18�24. 14. Cohen, C., �Maximum Likelihood Estimation in the Weibull Distribution Based on Complete and on Censored

Samples,� Technometrics, vol. 7, no. 4, Nov 1965, pp. 579�588. 15. Harter, L. and Moore, A., �Maximum-Likelihood Estimation of the Parameters of Gamma and Weibull

Populations from Complete and from Censored Samples,� Technometrics, vol. 7, no. 4, Nov. 1965, pp. 639�643.

16. McCool, J., �Inference on Weibull Percentiles and Shape Parameters from Maximum Likelihood Estimates,� IEEE Transactions on Reliability, vol. R�19, no. 1, Feb. 1970, pp. 2�9.

17. McCool, J., �Statistical Methods to Analyze Endurance� Ch. 20 in Rolling Bearing Analysis 3rd edition, T.A. Harris, editor, John Wiley and Sons, NY, 1992.

18. Houpert, L., �An Engineering Approach to Confidence Intervals and Endurance Test Strategies,� Tribology Transactions, vol. 46, no. 2, 2003, pp. 248�259.

19. Poplawski, J.V., Peters, S.M., and Zaretsky, E.V., �Effect of Roller Profile on Cylindrical Roller Bearing Life Prediction-Part I: Comparison of Bearing Life Theories,� STLE Tribology Trans., 44, 3, 2001, pp. 339�350.

20. Poplawski, J.V., Peters, S.M., and Zaretsky, E.V., �Effect of Roller Profile on Cylindrical Roller Bearing Life Prediction-Part II, Comparison of Roller Profiles,� STLE Tribology Trans., 44, 3, 2001, pp. 417�427.

NASA/TM�2003-212186 23

APPENDIX G. Lundberg and A. Palmgren (5) using the Weibull equation (2�4) first derived the relationship between

individual component lives and system life where

[ ] [ ]∑=

=n

i

ei

esys

1

/1/1 LL (3)

Therefore, the system life of a bearing composed of three components (inner race, rolling element, and outer race) that can fail is expressed as

[1/Lsys]

e = [1/Lir]e + [1/Lre]

e + [1/Lor]e Eq. (5)

where Lir is the life of the inner race, Lre is the life of the rolling element, Lor is the life of the outer race and e is the Weibull slope. The fraction of failures due to the failure of a component of a system is expressed by Johnson (11) as

e

ir

sys

L

Lfailuresiroffraction

= Eq. (A�1a)

e

re

sys

L

Lfailuresreoffraction

= Eq. (A�1b)

e

or

sys

L

Lfailuresoroffraction

= Eq. (A�1c)

Therefore, to determine the fraction of inner-ring (ir) failures Eq. (5), which is a definition of the total system

life, is multiplied by Lire.

[ ] [ ]

++=

eor

ere

eir

eire

sys

eir

LLLL

LL

1111 Eq. (A�2)

Thus, from Eqs. (A�1a) and (A�2)

e

or

ire

re

ire

or

ire

re

ire

ir

irir

sys

L

L

L

L

L

L

L

L

L

L

e

L

Lfailuresiroffraction

+

+

=

+

+

=

=

1

11 Eq. (A�3a)

Subsequently, the fraction of rolling-element (re) failures can be expressed as

e

or

ree

ir

rere

sys

L

L

L

L

e

L

Lfailuresreoffraction

++

=

=

1

1 Eq. (A�3b)

and the fraction of outer-ring (or) failures can be expressed as

1

1

+

+

=

=

e

re

ore

ir

oror

sys

L

L

L

L

e

L

Lfailuresoroffraction Eq. (A�3c)

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Available electronically at http://gltrs.grc.nasa.gov

August 2003

NASA TM—2003-212186

E–13789

WBS–22–323–71–00

29

Determination of Rolling-Element Fatigue Life From Computer GeneratedBearing Tests

Brian L. Vlcek, Robert C. Hendricks, and Erwin V. Zaretsky

Rolling-element bearings; Life prediction; Weibull analysis; Computer simulation;Monte Carlo analysis

Unclassified -UnlimitedSubject Categories: 37 and 39

Prepared for the 2003 Annual Meeting and Exhibition sponsored by the Society of Tribologists and LubricationEngineers, New York City, New York, April 28–May 1, 2003. Brian L. Vlcek, Georgia Southern University, Statesboro,Georgia 30460; Robert C. Hendricks and Erwin V. Zaretsky, NASA Glenn Research Center. Responsible person,Erwin V. Zaretsky, organization code 5900, 216–433–3241.

Two types of rolling-element bearings representing radial loaded and thrust loaded bearings were used for this study.Three hundred forty (340) virtual bearing sets totaling 31400 bearings were randomly assembled and tested by MonteCarlo (random) number generation. The Monte Carlo results were compared with endurance data from 51 bearing setscomprising 5321 bearings. A simple algebraic relation was established for the upper and lower L10 life limits as functionof number of bearings failed for any bearing geometry. There is a fifty percent (50 percent) probability that the resultantbearing life will be less than that calculated. The maximum and minimum variation between the bearing resultant lifeand the calculated life correlate with the 90-percent confidence limits for a Weibull slope of 1.5. The calculated lives forbearings using a load-life exponent p of 4 for ball bearings and 5 for roller bearings correlated with the Monte Carlogenerated bearing lives and the bearing data. STLE life factors for bearing steel and processing provide a reasonableaccounting for differences between bearing life data and calculated life. Variations in Weibull slope from the MonteCarlo testing and bearing data correlated. There was excellent agreement between percent of individual componentsfailed from Monte Carlo simulation and that predicted.

Distribution: Nonstandard

determination of rolling-element fatigue life from ... of rolling-element fatigue life ... virtual...

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